Conjecture on the mass curve of cubic-quintic ground states

Let $P_\omega$ denote the ground state soliton for the cubic-quintic nonlinear Schrödinger equation on $\mathbb{R}^3$, and let $M(P_\omega)$ be its mass. Mass-curve conjecture. There is an $\omega_*\in(0,\tfrac3{16})$ such that the map $\omega\mapsto M(P_\omega)$ is strictly decreasing for $\omega<\omega_*$ and strictly increasing for $\omega>\omega_*$. This would describe the numerically observed single turning point of the ground-state mass/energy curve; the source presents it as an open question.